226 problems

Fix an integer $d\geq 1$. For an unordered partition $(d_1,\ldots,d_r)\vdash d$ into strictly positive parts, let $\#\operatorname{Aut}(d_1,\ldots,d_r)$ be the order of its automor…

The big quantum cohomology of G/P is generically semisimple.

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singula…

Let $X$ be the smooth projective variety under consideration, let $N^{\operatorname{GW}}_{g,\beta}(X)$ denote its Gromov–Witten invariants, let $u$ be a formal variable, and let…

Let $h$ be the hyperplane class of $CP^2$, and let $w({\cal O}_{h^a}{\cal O}_{h^b}\mid(x^0,x^1,x^2))_0$ denote the generating function of multi-point virtual structure constants. D…

Let $X_1$ be the resolved conifold $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)$ equipped with the anti-diagonal action, and consider its equivariantly Calab…

Let $X$ be a smooth Fano variety of complex dimension $r$, let $g$ be a genus, and let $A$ be a curve class. The fixed-domain Gromov–Witten invariants are the virtual counts associ…

Multiple cover formula.

Crepant resolution conjecture. The transformation $\mathds{T}$ satisfies

Let $r_{g,h}$ denote the BPS count associated to a class $\beta$ on a K3 surface whose norm square satisfies … Let $q$ and $y$ be formal variables. Katz–Klemm–Vafa product formula.…

Klemm–Pandharipande integrality conjecture.

Let $(Q,G)$ be an LG pair, let $\mathcal{Z}=\{Q=0\}\subset\mathbb{P}(G)$ be the associated Calabi–Yau variety, and let $\mathscr{L}^{(Q,G)}\subset\mathscr{V}^{(Q,G)}$ and…

Let $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated gauge group $\mathsf{G}$, representation space $\mathsf{N}$, stability con…

Let $P(a_1,\dots,a_N\mid k_1,\dots,k_m)$ be a nonsingular complete intersection in weighted projective space. Let $t^p(x^*)$ be the mirror maps defined from the genus-zero virtual…

Relative BPS integrality conjecture. For every $d\geq 1$,

Consider the Lagrangian cones $\mathcal L_{\operatorname{FJRW}}$ and $\mathcal L_{\operatorname{GW}}$ in the symplectic spaces $\mathcal V_{\operatorname{FJRW}}$ and…

Let $X$ be a Calabi–Yau $3$-fold. Define the Gromov–Witten generating series by … and the reduced Donaldson–Thomas series by…

Let $M_N^k$ be the degree-$k$ projective Fano hypersurface considered in the paper, and let $\langle {\cal O}_{h^a}{\cal O}_{h^b}\rangle_0(t^0,t^1,\ldots,t^{N-2})$ denote the pertu…

Let $M$ be a Calabi–Yau threefold, and let $\beta$ be a nonzero curve class in $M$. The symbols $\operatorname{Z_{GW}^{\prime}}(M;u)_\beta$ and $\operatorname{Z_{PT}}(M;q)_\beta$ d…

Let $\mathcal{X}$ be an orbifold with projective coarse moduli space $X$, and let $\pi:Y\to X$ be a crepant resolution. Suppose that the big quantum products $\bullet_\tau$ for…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the diagonal torus action $\lambda_1=\lambda_2$, and let…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the antidiagonal torus action $(\lambda,-\lambda)$ on the bundle, and let $t_k$ denote the Kähler class o…

Let $\sum_i c_i\Gamma_i=0$ be a tautological equation in codimension $k$ on $\overline{M}_{g,n}$, and let $\mathfrak{r}_l$ be the operators sending decorated graphs on…

Let $\pi\colon\mathcal W\to\mathbb P^{1}$ be a family of quintic mirror threefolds. Let $N_g(d)$ be the genus-$g$ Gromov–Witten invariant of degree $d$ of a general quintic threefo…

Let $Y$ be a variety with transversal $A_n$-singularities, $n\geq 2$, and let $\rho:Z\to Y$ be its crepant resolution. Let $K$ be the line bundle on the singular locus appearing in…